Risk is the potential of experiencing a loss when a system does not operate as expected due to uncertainties. Its assessment requires the quantification of both the system failure potential and the multi-faceted failure consequences, which affect further systems. Modern industries (including the engineering and financial sectors) require increasingly large and complex models to quantify risks that are not confined to single disciplines but cross into possibly several other areas. Disasters such as hurricane Katrina, the Fukushima nuclear incident and the global financial crisis show how failures in technical and management systems cause consequences and further failures in technological, environmental, financial, and social systems, which are all inter-related. This requires a comprehensive multi-disciplinary understanding of all aspects of uncertainty and risk and measures for risk management, reduction, control and mitigation as well as skills in applying the necessary mathematical, modelling and computational tools for risk oriented decision-making. This complexity has to be considered in very early planning stages, for example, for the realisation of green energy or nuclear power concepts and systems, where benefits and risks have to be considered from various angles. The involved parties include engineering and energy companies, banks, insurance and re-insurance companies, state and local governments, environmental agencies, the society both locally and globally, construction companies, service and maintenance industries, emergency services, etc. The CDT is focussed on training a new generation of highly-skilled graduates in this particular area of engineering, mathematics and the environmental sciences based at the Liverpool Institute for Risk and Uncertainty. New challenges will be addressed using emerging probabilistic technologies together with generalised uncertainty models, simulation techniques, algorithms and large-scale computing power. Skills required will be centred in the application of mathematics in areas of engineering, economics, financial mathematics, and psychology/social science, to reflect the complexity and inter-relationship of real world systems. The CDT addresses these needs with multi-disciplinary training and skills development on a common mathematical platform with associated computational tools tailored to user requirements. The centre reflects this concept with three major components: (1) Development and enhancement of mathematical and computational skills; (2) Customisation and implementation of models, tools and techniques according to user requirements; and (3) Industrial and overseas university placements to ensure industrial and academic impact of the research. This will develop graduates with solid mathematical skills applied on a systems level, who can translate numerical results into languages of engineering and other disciplines to influence end-users including policy makers. Existing technologies for the quantification and management of uncertainties and risks have yet to achieve their significant potential benefit for industry. Industrial implementation is presently held back because of a lack of multidisciplinary training and application. The Centre addresses this problem directly to realise a significant step forward, producing a culture change in quantification and management of risk and uncertainty technically as well as educationally through the cohort approach to PGR training.

Gas-liquid foams are ubiquitous in our daily life and in industry. Applications range from food, consumer goods, pharmaceuticals, polymers and ceramics to fire-fighting, enhanced oil recovery, and mineral particle transport. Recently, applications have also emerged in the medical field, e.g. foam sclerotherapy of varicose veins, and expanding polymer foam for treating brain aneurysms. Thus, foams are crucial to a wide range of industries and contribute considerably to the world economy. For example, by 2018 the global market will be worth $61.9 billion for polyurethane foam, $7.9 billion for shaving foam, and $74 billion for ice cream. The chocolate market will reach $98.3 billion in 2016, and a considerable part of it is due to aerated products (e.g. mousse). Foams are challenging complex fluids which are used for a variety of reasons including their light weight, complex microstructure, rheology, and transience, many aspects of which are not well understood and, thus, not well predicted by current models. A wide gap therefore exists between the complexity of foam phenomena and the present state of knowledge, which makes foam design and control in commercial applications more art than science. In particular, in many industrial processes foams are forced to flow through intricate passages, into vessels with narrow complex cross-sections or through nozzles. Examples include flow of aerated confectionary in narrow channels and complex moulds, filling of cavities with insulation foam, flow of foamed cement slurries in narrow oil-well annuli, filling of hollow aerofoil sections with polyurethane foam to make aerodynamic tethers for communication and geoengineering applications, and production of pre-insulated pipes for district heating. These flows are typified by contractions and expansions which generate complex phenomena that can have important effects on foam structure and flow, and can lead to dramatic instabilities and morphological transformations with serious practical implications for foam sustainability during flow and processing. Here, the flow characteristics of the foam at bubble scale are important, but the topological changes incurred and their effects on the rheology and flow of the foam are poorly understood. This proposal seeks to address this lack of understanding by studying experimentally, using a range of advanced diagnostic techniques, and via theory and computer simulation a number of fundamental aspects related to the flow, stability and behaviour of three-dimensional foams through narrow channels containing a variety of complex geometries. The flow of aqueous foams as well as setting polymer foams with formulations of varying degrees of complexity will be experimentally studied. We will develop bubble-scale simulations with arbitrary liquid fractions spanning the whole range from dry to wet, to cover foams of industrial relevance. The wide range of experimental information and data to be generated in this project will allow these simulations to be guided and critically tested and, conversely, the simulations will underpin our engineering theory of the behaviour of foam flows in complex geometries. This basic knowledge, from theory, modelling and experiment, will give a step improvement in fundamental science, and will assist designers and manufacturers of foam products, as well as designers and users of foam generating or processing equipment. More specifically, the practical aim of the project is to develop predictive tools as an aid to industrial practitioners, to describe the structural and dynamical properties of foams in terms of formulation properties and flow parameters, based on the knowledge gained from the experimental and modelling work. We will also work with our industrial partners to help them improve their understanding of the fundamental science which underpins their particular foam flow applications and, thus, enable them to enhance them.

The principal aim of the research proposal is to develop a next generation multi-phase flow instrument to non-invasively measure the phase flow rates, and rapidly image the flow-field distributions, of complex, unsteady two- or three-phase flows. The proposed research is multi-disciplinary covering aspects of fluid mechanics modelling, sensor material selection and flow metering, process tomography and multi-variable data fusion. The new instrument will be based on the novel concepts of 3D vector Electrical Impedance Tomography (EIT) and the Electromagnetic Velocity Profiler (EVP). These will be used in conjunction with auxiliary differential-pressure measurements for flow density and total flow rate. It is our intention to be able to measure the volumetric flow rate, image time-dependent distributions of the local axial velocity and volume fraction of the dispersed and continuous phases, visualise flow patterns and provide an alternative measurement of volumetric flow rates in two and three phase flows. The project draws upon several recent advances in EIT technology made by the proposers' research teams. Together these potentially enable the development of an advanced flow meter intended to address some limitations of current multiphase flow meters, leading to improvements of the management of productivity in many industrial sectors such as petroleum, petrochemical, food, nuclear and mineral processing. Within the scope of this research, only flows with a conductive continuous liquid phase will be targeted. We will make use of advanced Magnetic Resonance Imaging (MRI) protocols for independent non-invasive validation of both the phase volume fraction and velocity distribution measurements. It is intended that the project will pave the way for the manufacturing of a next generation of advanced multi-phase flow measurement and rapid visualisation technologies.

This project is concerned with developing and prototyping new, more effective computer simulation methods for modelling the propagation and reflection of seismic waves in the earth subsurface. The pull for this research comes from the CASE partner, Schlumberger Cambridge Research, who wish to develop more efficient methods for imaging the subsurface below land or marine deposits in order to locate hydrocarbon-bearing rocks. In particular, for accurate imaging in more complex geometries where seismic reflectivity is poor, for instance imaging below basalt or salt structures, the state-of the art is to use imaging methods which proceed iteratively, solving the full mathematical equations describing the wave propagation and reflection at each step. Unfortunately, this full simulation of seismic propagation and reflection by standard numerical methods is hugely expensive in computing resources, because of the complex subsurface geometry and the large 3D region that must be simulated. Large, that is, in diameter in comparison with the wavelengths of the seismic waves, so that a very high resolution is needed to visualise the wave propagation accurately using standard computational methods. These standard methods include so-called 'finite element methods', computer simulation methods in which the earth subsurface is thought of as composed of a large number (e.g. 1,000,000-100,000,000) of small pieces (the 'finite elements') in each of which the seismic wave has a very simple behaviour, e.g. is approximately constant. This project is concerned with exploring the use, for seismic simulations, of a new, more sophisticated class of finite element method. This new class of method differs in using a more sophisticated assumed behaviour in each element, namely a certain standard wave-like behaviour (that of a plane wave or a combination of plane waves). To keep the project to a manageable size, one suitable for proof of concept, and suitable for a PhD student to complete in 3 1/2 years, the modelling will be restricted to two-dimensional simulations (where it is assumed that the geometry is constant in one horizontal direction), and to a simplified acoustic model of the seismic propagation. Main objectives of the project will be: i) To extend previous work of this type, namely the so-called Ultra Weak Variational Formulation, so that the method can deal with spatially varying seismic properties (that is, where the wave speed varies gradually or suddenly with position in the subsurface). This (significant) extension to the current method, which will need both strong mathematical and computing skills, will be essential for the method to be of use for general purpose seismic modelling. ii) To apply a combination of sophisticated mathematics and numerical experiments so as to understand the behaviour of the new algorithm. iii) To test the new method on representative acoustic 2D geological models supplied by Schlumberger, comparing the performance of the new algorithms, as implemented in computer software, with existing methods, based on standard finite elements and so-called finite difference modelling. These standard methods Schlumberger has implemented in existing computer software. In the first 18 months of the PhD the student will receive training, in superb research environments at Reading and Schlumberger, in the mathematics, computing, and knowledge of standard methods for modelling seismic propagation, that will be necessary for completion of the project. In this, the student at Reading will benefit from access to courses forming part of our MSc in Mathematics of Scientific and Industrial Computation, from our membership of an advanced graduate level training consortium in Mathematics (the MAGIC group), and from membership of a large community of academic and research staff and PhD students working on many exciting applications of the mathematics of waves, and the use of waves in imaging.

Geometry and number theory are core disciplines within pure mathematics, with many repercussions across science and society. They are subjects that have attracted some of the best minds in mathematics since the time of the Ancient Greeks and continue to exert a natural fascination on professional and amateur mathematicians alike. Throughout the history of mathematics, both topics have very often inspired major mathematical developments which have had enormous impact beyond their original applications. The fascination of number theory is exemplified by the story of Fermat's last theorem, the statement of which was written down in 1637 and which is simple enough to be understood by anyone familiar with high school mathematics. It took more than 350 years of hard work and highly significant developments across mathematics before Wiles's celebrated proof was finally published in 1995. Wiles's proof involves a mixture of ideas from number theory and geometry, and the interplay between these topics is one of the most active areas of research in pure mathematics today. The Centre is needed to educate the next generation of academic researchers to maintain the excellence and competitiveness of the UK's universities and also to deliver highly trained mathematicians ready to take their place in financial and other high-tech industries. As shown by our letters of support from the Bank of England, the Satellite Applications Catapult, Heilbronn, Royal Bank of Scotland, and Schlumberger, a wide range of employers have the vision to invest in highly trained pure mathematicians. Our partners all speak highly of the analytical and problem-solving abilities of pure mathematicians trained to PhD level. The students trained in this Centre will be even more highly skilled: the structure of the training programme will encourage independence and leadership and will embed professional development and key skills such as programming, communication skills and public engagement alongside cutting-edge research in topics chosen from geometry and number theory.